$\dfrac{ 8i + 9j }{ -6 } = \dfrac{ 5i - 10k }{ 7 }$ Solve for $i$.
Multiply both sides by the left denominator. $\dfrac{ 8i + 9j }{ -{6} } = \dfrac{ 5i - 10k }{ 7 }$ $-{6} \cdot \dfrac{ 8i + 9j }{ -{6} } = -{6} \cdot \dfrac{ 5i - 10k }{ 7 }$ $8i + 9j = -{6} \cdot \dfrac { 5i - 10k }{ 7 }$ Multiply both sides by the right denominator. $8i + 9j = -6 \cdot \dfrac{ 5i - 10k }{ {7} }$ ${7} \cdot \left( 8i + 9j \right) = {7} \cdot -6 \cdot \dfrac{ 5i - 10k }{ {7} }$ ${7} \cdot \left( 8i + 9j \right) = -6 \cdot \left( 5i - 10k \right)$ Distribute both sides ${7} \cdot \left( 8i + 9j \right) = -{6} \cdot \left( 5i - 10k \right)$ ${56}i + {63}j = -{30}i + {60}k$ Combine $i$ terms on the left. ${56i} + 63j = -{30i} + 60k$ ${86i} + 63j = 60k$ Move the $j$ term to the right. $86i + {63j} = 60k$ $86i = 60k - {63j}$ Isolate $i$ by dividing both sides by its coefficient. ${86}i = 60k - 63j$ $i = \dfrac{ 60k - 63j }{ {86} }$